Question

What is the standard form of the equation of the parabola with the focus (−1,3) and the directrix x=9?

Select the correct answer below:

a) (x+1)2=−12(y−6)

b) (y−3)2=−20(x−4)

c) (x−9)2=8(y−1)

d)(y−3)2=20(x−4)

please show work and what is the correct answer? thanks

Answer 1

Let (x, y) be any point on the parabola.

Then the distance between the focus of the parabola and the point (x, y) must be equal to the distance between the the directrix and the point (x, y).

.

Now we are given that, focus = (-1, 3) and directrix is x = 9

.

So distance between (x, y) and focus is,

√[(x + 1)² + (y - 3)²]

And distance between (x, y) and the directrix is,

|x - 9|

.

So we must have,

√[(x + 1)² + (y - 3)²] = |x - 9|

Now by squaring both sides, we get,

(x + 1)² + (y - 3)² = (x - 9)²​​​​​​

i.e. x² + 2x + 1 + (y - 3)² = x² - 18x + 81

i.e. 2x + 1 + (y - 3)² = -18x + 81

i.e. (y - 3)² = -18x + 81 - 2x - 1

i.e. (y - 3)² = -20x + 80

i.e. (y - 3)² = -20(x - 4)

.

So option B is correct.